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Math Help - Identities Applications

  1. #1
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    Identities Applications

    Hello,

    I'm having trouble with a trigonometry problem, outlined below:

    cos²(2x) - sin²(2x) = (√3/2)

    I'm not quire sure how to solve for x, can any of you help?

    I would appreciate your assistance!

    Thanks
    Last edited by ElectroNerd; April 15th 2011 at 03:22 PM.
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  2. #2
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    Recall the identity cos²x-sin²x = cos(2x).
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  3. #3
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    I still don't quite understand the relationship between the identity you suggested:

    cos²x - sin²x = cos(2x)

    and the left term in the equation I have:

    cos²(2x) - sin²(2x)

    I'm not quite sure of what to do because of the '2x', otherwise I could easily substitute the identity you suggested.

    EDIT:

    I see now that in my case it would be cos(4x) as the substitution.
    Last edited by ElectroNerd; April 15th 2011 at 04:11 PM.
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  4. #4
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    Okay, I've determined that the following is true:

    cos(4x) = (√3/2)

    How do I simplify further?
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  5. #5
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    Quote Originally Posted by ElectroNerd View Post
    I see now that in my case it would be cos(4x) as the substitution.
    Yes, and (√3/2) = cos(π/6). So you've cos(4x) = cos(π/6), which implies 4x = π/6. Can you find the other solutions from that?
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  6. #6
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    Okay, I didn't realize that other identity!

    So I solved for x:

    x = π/24

    Just so I can understand how to solve more problems, how do you go about this? Do you have all the identities memorized? If I knew that cos(4x) = cos(π/6), then I could have solved this more easily.

    I guess I should just memorize all the identities, is that the best way?
    Last edited by ElectroNerd; April 15th 2011 at 05:16 PM.
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  7. #7
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    Quote Originally Posted by ElectroNerd View Post
    So I solved for x: x = π/24
    Right! That's what one solution, though. There are infinitely many. Can you find (or are you required to) find the rest?
    Just so I can understand how to solve more problems, how do you go about this? Do you have all the identities memorized? If I knew that cos(4x) = cos(π/6), then I could have solved this more easily.
    No, cos(4x) = cos(π/6) is not an identity. Our equation was cos(4x) = (√3/2), but (√3/2) = cos(π/6) so we have cos(4x) = cos(π/6).

    I guess I should just memorize all the identities, is that the best way?
    Memorisation will only get you so far; the best way is doing as many problems as you can get hold of.
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  8. #8
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    That is the answer they give, thanks!

    Do you know of any good trigonometry references I can study?
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  9. #9
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    Quote Originally Posted by ElectroNerd View Post
    Do you know of any good trigonometry references I can study?
    A brilliant book is this one: Advanced Trigonometry - C. V. Durell and A. Robson. It's very old, so you can probably download it online for free too.
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